Textbook Question
For the circuit of FIGURE EX32.32, What is the resonance frequency, in both rad/s and Hz?
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Ch 32: AC Circuits
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796
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Table of contents
- Ch 01: Concepts of Motion35
- Ch 02: Kinematics in One Dimension75
- Ch 03: Vectors and Coordinate Systems58
- Ch 04: Kinematics in Two Dimensions60
- Ch 05: Force and Motion23
- Ch 06: Dynamics I: Motion Along a Line53
- Ch 07: Newton's Third Law27
- Ch 08: Dynamics II: Motion in a Plane52
- Ch 09: Work and Kinetic Energy56
- Ch 10: Interactions and Potential Energy50
- Ch 11: Impulse and Momentum56
- Ch 12: Rotation of a Rigid Body71
- Ch 13: Newton's Theory of Gravity52
- Ch 14: Fluids and Elasticity44
- Ch 15: Oscillations55
- Ch 16: Traveling Waves37
- Ch 17: Superposition39
- Ch 18: A Macroscopic Description of Matter53
- Ch 19: Work, Heat, and the First Law of Thermodynamics60
- Ch 20: The Micro/Macro Connection53
- Ch 21: Heat Engines and Refrigerators34
- Ch 22: Electric Charges and Forces40
- Ch 23: The Electric Field39
- Ch 24: Gauss' Law32
- Ch 25: The Electric Potential63
- Ch 26: Potential and Field57
- Ch 27: Current and Resistance42
- Ch 28: Fundamentals of Circuits39
- Ch 29: The Magnetic Field47
- Ch 30: Electromagnetic Induction60
- Ch 31: Electromagnetic Fields and Waves32
- Ch 32: AC Circuits63
- Ch 33: Wave Optics32
- Ch 34: Ray Optics57
- Ch 35: Optical Instruments30
- Ch 36: Special Relativity38
- Ch 37: The Foundations of Modern Physics25
- Ch 38: Quantization49
- Ch 39: Wave Functions and Uncertainty31
- Ch 40: One-Dimensional Quantum Mechanics35
- Ch 41: Atomic Physics18
- Ch 42: Nuclear Physics27
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 32, Problem 28
A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled and, at the same time, the inductor value is halved?
Verified step by step guidance1
Step 1: Recall the formula for the resonance frequency of an RLC circuit: \( f_r = \frac{1}{2\pi\sqrt{LC}} \), where \( L \) is the inductance and \( C \) is the capacitance.
Step 2: Identify the initial resonance frequency \( f_r \) given in the problem as 200 kHz, and note the changes to the circuit components: the capacitance \( C \) is doubled (\( C' = 2C \)) and the inductance \( L \) is halved (\( L' = \frac{L}{2} \)).
Step 3: Substitute the new values of \( L' \) and \( C' \) into the resonance frequency formula: \( f_r' = \frac{1}{2\pi\sqrt{L'C'}} = \frac{1}{2\pi\sqrt{\frac{L}{2} \cdot 2C}} \).
Step 4: Simplify the expression inside the square root: \( \sqrt{\frac{L}{2} \cdot 2C} = \sqrt{LC} \). This shows that the resonance frequency remains unchanged.
Step 5: Conclude that the resonance frequency of the circuit is still 200 kHz, as the changes to \( L \) and \( C \) cancel each other out in the formula.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Resonance Frequency in RLC Circuits
The resonance frequency of a series RLC circuit is the frequency at which the inductive reactance equals the capacitive reactance, resulting in maximum current flow. It is given by the formula f₀ = 1 / (2π√(LC)), where L is the inductance and C is the capacitance. At resonance, the impedance of the circuit is minimized, and the circuit can oscillate freely.
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Resonance in Series LRC Circuits
Effect of Capacitance on Resonance Frequency
Doubling the capacitance in an RLC circuit affects the resonance frequency inversely. According to the resonance frequency formula, increasing capacitance decreases the resonance frequency, as it is in the denominator of the square root. This means that if the capacitance is doubled, the resonance frequency will decrease, leading to a lower frequency of oscillation.
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05:23Resonance in Series LRC Circuits
Effect of Inductance on Resonance Frequency
Halving the inductance in an RLC circuit also influences the resonance frequency, but in the opposite direction. Since inductance is also in the denominator of the square root in the resonance frequency formula, reducing the inductance increases the resonance frequency. Therefore, if the inductance is halved, the resonance frequency will increase, resulting in a higher frequency of oscillation.
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Related Practice
Textbook Question
For the circuit of FIGURE EX32.32, Find VR and VL at resonance.
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Textbook Question
A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled?
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Textbook Question
An inductor is connected to a 15 kHz oscillator. The peak current is 65 mA when the rms voltage is 6.0 V. What is the value of the inductance L?
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Textbook Question
For the circuit of FIGURE EX32.33, What is the resonance frequency, in both rad/s and Hz?
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Textbook Question
A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the resistor value is doubled?
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